The Fresnel Equations and Brewster's Law
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The Fresnel Equations and Brewster's Law Equipment Optical bench pivot, two 1 meter optical benches, green laser at 543.5 nm, 2 10cm diameter polarizers, rectangular polarizer, LX-02 photo-detector in optical mount, thick acrylic block, thick glass block, Phillips multimeter, laser mount, sunglasses. Purpose To investigate polarization by reflection. To understand and verify the Fresnel equations. To explore Brewster’s Law and find Brewster’s angle experimentally. To use Brewster’s law to find Brewster’s angle. To gain experience working with optical equipment. Theory Light is an electromagnetic wave, of which fundamental characteristics can be described in terms of the electric field intensity. For light traveling along the z-axis, this can be written as r r i(kz−ωt) E = E0e (1) r where E0 is a constant complex vector, and k and ω are the wave number and frequency respectively, with k = 2π / λ , (2) λ being the wavelength. The purpose of this lab is to explore the properties the electric field in (1) at the interface between two media with indices of refraction ni and nt . In general, there will be an incident, reflected and transmitted wave (figure 1), which in certain cases reduce to incident and reflected or incident and transmitted only. Recall that the angles of the transmitted and reflected beams are described by the law of reflection and Snell’s law. This however tells us nothing about the amplitudes of the reflected and transmitted Figure 1 electric fields. These latter properties are defined by the Fresnel equations, which we review below. First note that any plane wave may be represented as a superposition of two orthogonal linearly polarized waves. By decomposing the electric field into linearly polarized light parallel and perpendicular to the plane of incidence, we can apply the boundary conditions at the interface for each case. The components of the electric and magnetic fields which are tangent to the surface must be continuous across the boundary. We begin by concentrating on the case shown in figure 1, i.e. the light wave polarized in the plane of incidence1. For the electric field the continuity relation becomes 1 By convention, the polarization direction of a light wave that of the electric (not magnetic) field vector. − Ei cosθi + Er cosθi = −Et cosθt (3) and for the magnetic field B B B i + r = t (4) µ1 µ1 µ2 where the law of reflection has been used to set θ r = θi and we took into account that the r r r r r r vectors E, B,k are mutually orthogonal with E × B oriented along k . Here, µ is the permeability of the material. Further taking into account the fact that in an electromagnetic wave and nα Eα ∝ Bα (with α = i,r,t and nα being the refraction index of the material through which the corresponding wave is propagating) we can write equation (4) as: ni nt (Ei + Er ) = Et . (5) µi µt Combining with equation (3), we arrive at the Fresnel equations for the wave polarized in the plane of incidence: nt cosθi − ni cosθt r|| = (6) nt cosθi + ni cosθt 2ni cosθi t|| = , (7) nt cosθi + ni cosθt where we’ve supposed that we are in a non-magnetic material ( µ = 1 ) and defined the amplitude reflection/transmission coefficients: E r = r (8) Ei E t = t (9) Ei as the ratios of the amplitude of the reflected electric field to the amplitude of the incident electric field. In the case that the electric field is parallel to the plane of incidence, in much the same way, we derive the second pair of Fresnel equations: ni cosθi − nt cosθt r⊥ = (10) ni cosθi + nt cosθt 2ni cosθi t⊥ = (11) ni cosθi + nt cosθt Equations (6) and (10) may be conveniently expressed using Snell’s law where they become: sin(θi −θt ) r⊥ = − (12) sin(θi +θt ) tan(θi −θt ) r|| = (13) tan(θi +θt ) The reflectance (intensity reflection coefficient) is the square of the amplitude reflection coefficient: R = r 2 . (14) The last three equations combine to give: 2 sin (θi −θt ) (15) R⊥ = 2 sin (θi +θt ) tan 2 (θ −θ ) i t . (16) R|| = 2 tan (θi +θt ) Consider now the case in which light is traveling from a medium of lower to higher index of refraction (say, an air/glass interface). Since in this case, θi > θt , (15) is never zero. However, the right π hand side of equation (16) vanishes when θ +θ = i t 2 in which case the denominator becomes infinite. At this point, all incident light which is polarized in the plane tangent to the plane of incidence is transmitted. The value of θi at which this occurs is known as Brewster’s Angle θ B . Writing Snell’s law at Brewster’s angle, π n sinθ = n sinθ = n sin( −θ ) = n cosθ , i B t t t 2 B t B or nt tanθB = (17) ni which is Brewster’s Law2. 2 Sir David Brewster (1781-1868) is mostly remembered for his invention of the kaleidoscope and for optical improvements of the microscope. However, his main experiments were on the theory of light and its uses. His first paper, "Some Properties of Light", was published in 1813. Brewster's Law was named after him in 1814 when he made measurements on the angle of maximum polarization using biaxial crystals. He was awarded all three of the principal medals of the Royal Society for his optical research. These were the Copley medal in 1815, the Rumford medal in 1818, and the Royal medal in 1830. He was also knighted in 1831. Brewster’s Law may be understood by the following intuitive argument: the electric field vector is transverse to the forward direction. However, at the point where the light polarized in the plane of incidence starts to enter the glass the vibrations in the transmitted wave happen to be parallel to the OR ray in Figure 3. Elementary molecular dipoles inside the material also oscillate along OR, and cannot produce a “reflected” electromagnetic wave propagating in the same direction. This means that only the component polarized normal to the incident plane, SOR, or parallel to the surface, will be reflected. At this point, even if the incident wave has both polarization components (or its polarization its random), ray OR is completely horizontally polarized and the electric field vector is restricted to the plane that is parallel to the reflected surface. This 100% linear polarization can only occur at one angle, because this circumstance is restricted by the fact that the angle between the reflected ray, OR, and the refracted ray, OT, has to be 90o. A graph of the reflectivity (15)-(16) for the air-glass interface as a function of the angle of incidence is shown in Figure 4. At normal incidence, the parallel and perpendicular polarization waves are physically identical, and have the same reflectivity. Notice that even though the glass is transparent, 4% of the light is still reflected. However, as θi increases, the parallel component drops and the perpendicular component rises, until at the Brewster's angle their values are approximately 0% and 15% of the initial intensity of the light respectively. At an incident angle of 90o, all of the incident light is reflected, so the substance acts as a mirror. The Fresnel equations can be tested by using a pivoting lab bench to vary the incident angle of laser light onto a reflecting surface and measuring the reflected intensity. A photodetector is used in this lab to measure the light intensity. The photodetector is calibrated to generate a voltage proportional to the light intensity, which is then measured with a multimeter. A polarizer between the dielectric surface and the photodetector is used to select the parallel and perpendicular components of the reflected light. Experimental Procedure 1. The theoretical index of refraction for the crown glass and acrylic used in this lab are 1.61 and 1.49 respectively. With these values, use Snell’s Law for θt , in equations (15) and (16) to make a theoretical plot of the Fresnel Equations similar to figure 4. 2. Throughout this experiment ensure that the apparatus is aligned so that both the reflected and transmitted laser beams are pointed away from the other students, and always keep track of where the beams are. The laser you will be using for this experiment will itself produce polarized light. Place the laser, polarizer, and detector onto the same optical bench. Turn on the laser, voltmeter and the photodetector. Make sure the voltmeter is set to mV setting, and that the detector is set to the 0.1mV = 1Lux setting. 3. The first task is to calibrate the polarizers spatially, and with respect to one another. Since in this experiment, you will need to measure light parallel and perpendicular to the plane of incidence which is determined by your set-up, it is important to know what value on the polarizers correspond to the ⊥ (vertical) and || (horizontal) planes. The rectangular polarizer has its plane of transmission along the longer side. Send the laser through one of the “round” polarizers and the rectangular sheet polarizer oriented vertically. Place a paper card behind both polarizers. Rotate the round polarizer until the transmission through both polarizers is minimized. Make a note of the rotation angle of your polarizer. This angle corresponds to a horizontally polarized wave. Repeat this procedure with the second “round” polarizer. 4. Set up the bench as in figure 4 with the glass slider in the holder. Rotate polarizer 1 to transmit 45° polarization.